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Thread: A polynomial graph, with equal or mixed or what roots?!

  1. #1 A polynomial graph, with equal or mixed or what roots?! 
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    Hey!

    When we consider a polynomial, say cubic, we say that its graph will cut the x axis at atmost 3 points and the x co-ordinates of those points will the roots of the polynomial.

    But sometimes the graph of a cubic polynomial may cut the x axis at a single point, for example the graph of y = x^3 and this means that THIS POLYNOMIAL HAS 3 IDENTICAL ZEROES each equal to 0 {bcoz y = (x-0)(x-0)(x-0)}

    But is it the case every time? Well, if the graph of a cubic polynomial cuts the x axis at one point, can it NOT mean that

    1) the other two of its roots are complex and not real values (btw, that is true if we consider this: y = x^3 - 1)
    2) my real question is, can it happen that the other two roots do not exist at all? Not even in the complex world? So that we can say "the other two roots do not exist and hence it cuts the x axis once only......"

    So, cutting of the x axis just once by a cubic polynomial means basically what, if someone asks us what should we tell - is it 3 equal roots? Or 1 real and rest complex roots.....all these things we cannot discern from the graph alone and we need to look at the equation, right?? Such a graph means just that there will be one real root and not sure about the status of others i.e.being equal to it or being complex?


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    KJW
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    Quote Originally Posted by Gaurav(-26.7) View Post
    can it happen that the other two roots do not exist at all? Not even in the complex world?
    No. There will always be three (possibly equal) roots. See "fundamental theorem of algebra".


    Quote Originally Posted by Gaurav(-26.7) View Post
    So, cutting of the x axis just once by a cubic polynomial means basically what, if someone asks us what should we tell - is it 3 equal roots? Or 1 real and rest complex roots.....all these things we cannot discern from the graph alone
    From the graph, one can distinguish between three equal roots and only one real root by looking at the slope of the graph at the root. If there are three equal roots, the slope of the graph at the root will be zero. If the slope of the graph at the root is non-zero, there is only one real root.


    There are no paradoxes in relativity, just people's misunderstandings of it.
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  4. #3  
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    Oh, now I see. Thanks for the help, KJW!
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  5. #4  
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    Oh, now I see. Thanks for the help, KJW.
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  6. #5  
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    Yes, KJW is quite right, but the proof of the Fundamental Theorem of Arithmetic is fiendish - attempt it at peril to your sanity!

    Definition: A number is said to algebraic if it is the root of a polynomial with integer coefficients.

    Fact: The overwhelming majority of algebraic numbers are complex
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